Describing 3-faces in normal plane maps with minimum degree 4

In 1940, Lebesgue proved that every 3-polytope with minimum degree at least 4 contains a 3-face for which the set of degrees of its vertices is majorized by one of the following sequences: (4,4,∞),(4,5,19),(4,6,11),(4,7,9),(5,5,9),(5,6,7).(4,4,∞),(4,5,19),(4,6,11),(4,7,9),(5,5,9),(5,6,7).Borodin (2002) strengthened this to (4,4,∞)(4,4,∞), (4,5,17)(4,5,17), (4,6,11)(4,6,11), (4,7,8)(4,7,8), (5,5,8)(5,5,8), (5,6,6)(5,6,6).We obtain the following description of 3-faces in normal plane maps with minimum degree at least 4 (in particular, it holds for 3-polytopes) in which every parameter is best possible and is attained independently of the others: (4,4,∞),(4,5,14),(4,6,10),(4,7,7),(5,5,7),(5,6,6).(4,4,∞),(4,5,14),(4,6,10),(4,7,7),(5,5,7),(5,6,6).